Triangular probability

Let us name vertices clockwise $P_i$ , for $i = 0, 1 ... 6$ .

Then $P_1P_3P_5$ and $P_2P_4P_6$ are the two possible equilateral triangles.

And since there are $\binom{6}{3} = 20$ combinations of triangles, $\frac{2}{20} = 0.1$ .

This calculation is remarkably slow, probably because sympy does exact calculations. Obviously it's unnecessary to do this anyway.

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from sympy import *
from itertools import combinations

h=RegularPolygon(Point(0,0),1,6)

allTrianglesCount=0
equilateralTrianglesCount=0
for c in combinations(h.vertices,3):
    allTrianglesCount+=1
    angles=Triangle(*c).angles.values()
    equilateralTrianglesCount+=(angles[0]==angles[1]==angles[2])

print equilateralTrianglesCount, allTrianglesCount
print 1.*equilateralTrianglesCount/allTrianglesCount